Rationalising a denominator rewrites an exact expression into an equivalent form whose denominator is rational. The value must not change. The method works because numerator and denominator are multiplied by the same nonzero quantity.
1. A single surd denominator
For 5/√3, multiply by √3/√3:
5/√3 = 5√3/3.
The fraction √3/√3 equals 1, so the numerical value is unchanged.
2. Simplify before rationalising
Consider 6/√12. Since √12=2√3, first write 6/(2√3)=3/√3. Rationalising gives √3.
Starting with a simplified denominator often reduces arithmetic.
3. Binomial denominators need a conjugate
For 1/(2+√3), multiplying only by √3 does not remove every surd from the denominator. Instead use the conjugate 2−√3.
1/(2+√3) × (2−√3)/(2−√3) = (2−√3)/(4−3) = 2−√3.
4. Why conjugates work
(a+b)(a−b)=a²−b². When b is a surd term, squaring it can produce a rational quantity and the middle terms cancel.
For example, (5+2√2)(5−2√2)=25−8=17.
5. Worked example with a numerator
Simplify (3+√2)/(4−√2).
Multiply by (4+√2)/(4+√2). The denominator is 16−2=14. The numerator is 12+3√2+4√2+2=14+7√2.
Therefore the expression is (14+7√2)/14 = 1+√2/2.
6. Preserve brackets
When multiplying by a conjugate, the entire numerator must multiply by the entire conjugate. Missing one cross-term changes the expression rather than rewriting it.
7. Check equivalence numerically without replacing exact form
2−√3 is approximately 0.268. The original 1/(2+√3) is also approximately 0.268. A decimal check can expose a sign error while the exact form remains the final algebraic result.
8. Rationalising is representation change
The expressions 1/√2 and √2/2 represent the same number. One form may combine more conveniently with another fraction or exact trigonometric value.
The goal is not to declare the original value wrong. It is to choose a useful equivalent form.
9. Common mistakes
- Multiplying only the denominator.
- Using the same sign instead of the conjugate sign.
- Forgetting cross-terms in the numerator.
- Stopping before simplifying a common factor.
10. Practice
- Rationalise 4/√5.
- Rationalise 3/(1+√2).
- Rationalise 2/(3−√5).
- Simplify (1+√3)/(2+√3).
- Show that 1/(√3−√2)=√3+√2.
11. Answers
- 4√5/5.
- 3(1−√2)/(1−2)=3(√2−1).
- 2(3+√5)/(9−5)=(3+√5)/2.
- Multiply by 2−√3: numerator −1+√3, denominator 1, so √3−1.
- Multiply by √3+√2; denominator 3−2=1.
Continue with Surds in Additional Mathematics or return to the Additional Mathematics Hub.